Properties

Label 177.12.a.c.1.13
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(0\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.53597 q^{2} -243.000 q^{3} -1975.14 q^{4} -453.475 q^{5} +2074.24 q^{6} -60294.7 q^{7} +34341.4 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-8.53597 q^{2} -243.000 q^{3} -1975.14 q^{4} -453.475 q^{5} +2074.24 q^{6} -60294.7 q^{7} +34341.4 q^{8} +59049.0 q^{9} +3870.85 q^{10} +689766. q^{11} +479958. q^{12} +805536. q^{13} +514674. q^{14} +110194. q^{15} +3.75194e6 q^{16} -447375. q^{17} -504041. q^{18} +1.38162e6 q^{19} +895675. q^{20} +1.46516e7 q^{21} -5.88783e6 q^{22} -2.70756e7 q^{23} -8.34496e6 q^{24} -4.86225e7 q^{25} -6.87603e6 q^{26} -1.43489e7 q^{27} +1.19090e8 q^{28} -9.84356e7 q^{29} -940616. q^{30} -2.28946e8 q^{31} -1.02358e8 q^{32} -1.67613e8 q^{33} +3.81878e6 q^{34} +2.73421e7 q^{35} -1.16630e8 q^{36} -5.53325e6 q^{37} -1.17934e7 q^{38} -1.95745e8 q^{39} -1.55730e7 q^{40} +6.36122e8 q^{41} -1.25066e8 q^{42} -1.32430e9 q^{43} -1.36238e9 q^{44} -2.67772e7 q^{45} +2.31116e8 q^{46} +2.16637e9 q^{47} -9.11722e8 q^{48} +1.65813e9 q^{49} +4.15040e8 q^{50} +1.08712e8 q^{51} -1.59104e9 q^{52} +1.51378e8 q^{53} +1.22482e8 q^{54} -3.12792e8 q^{55} -2.07060e9 q^{56} -3.35733e8 q^{57} +8.40244e8 q^{58} -7.14924e8 q^{59} -2.17649e8 q^{60} -1.17893e10 q^{61} +1.95427e9 q^{62} -3.56034e9 q^{63} -6.81026e9 q^{64} -3.65290e8 q^{65} +1.43074e9 q^{66} -1.16751e10 q^{67} +8.83627e8 q^{68} +6.57937e9 q^{69} -2.33392e8 q^{70} +2.77195e10 q^{71} +2.02782e9 q^{72} -9.19919e9 q^{73} +4.72317e7 q^{74} +1.18153e10 q^{75} -2.72888e9 q^{76} -4.15893e10 q^{77} +1.67088e9 q^{78} -2.72056e10 q^{79} -1.70141e9 q^{80} +3.48678e9 q^{81} -5.42992e9 q^{82} +1.20624e10 q^{83} -2.89389e10 q^{84} +2.02873e8 q^{85} +1.13042e10 q^{86} +2.39199e10 q^{87} +2.36875e10 q^{88} +5.23015e10 q^{89} +2.28570e8 q^{90} -4.85695e10 q^{91} +5.34780e10 q^{92} +5.56338e10 q^{93} -1.84921e10 q^{94} -6.26528e8 q^{95} +2.48729e10 q^{96} +1.76629e10 q^{97} -1.41537e10 q^{98} +4.07300e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.53597 −0.188620 −0.0943101 0.995543i \(-0.530065\pi\)
−0.0943101 + 0.995543i \(0.530065\pi\)
\(3\) −243.000 −0.577350
\(4\) −1975.14 −0.964422
\(5\) −453.475 −0.0648960 −0.0324480 0.999473i \(-0.510330\pi\)
−0.0324480 + 0.999473i \(0.510330\pi\)
\(6\) 2074.24 0.108900
\(7\) −60294.7 −1.35594 −0.677969 0.735090i \(-0.737140\pi\)
−0.677969 + 0.735090i \(0.737140\pi\)
\(8\) 34341.4 0.370530
\(9\) 59049.0 0.333333
\(10\) 3870.85 0.0122407
\(11\) 689766. 1.29134 0.645672 0.763615i \(-0.276577\pi\)
0.645672 + 0.763615i \(0.276577\pi\)
\(12\) 479958. 0.556810
\(13\) 805536. 0.601723 0.300861 0.953668i \(-0.402726\pi\)
0.300861 + 0.953668i \(0.402726\pi\)
\(14\) 514674. 0.255757
\(15\) 110194. 0.0374677
\(16\) 3.75194e6 0.894533
\(17\) −447375. −0.0764192 −0.0382096 0.999270i \(-0.512165\pi\)
−0.0382096 + 0.999270i \(0.512165\pi\)
\(18\) −504041. −0.0628734
\(19\) 1.38162e6 0.128010 0.0640048 0.997950i \(-0.479613\pi\)
0.0640048 + 0.997950i \(0.479613\pi\)
\(20\) 895675. 0.0625872
\(21\) 1.46516e7 0.782851
\(22\) −5.88783e6 −0.243574
\(23\) −2.70756e7 −0.877152 −0.438576 0.898694i \(-0.644517\pi\)
−0.438576 + 0.898694i \(0.644517\pi\)
\(24\) −8.34496e6 −0.213925
\(25\) −4.86225e7 −0.995789
\(26\) −6.87603e6 −0.113497
\(27\) −1.43489e7 −0.192450
\(28\) 1.19090e8 1.30770
\(29\) −9.84356e7 −0.891176 −0.445588 0.895238i \(-0.647005\pi\)
−0.445588 + 0.895238i \(0.647005\pi\)
\(30\) −940616. −0.00706717
\(31\) −2.28946e8 −1.43629 −0.718147 0.695892i \(-0.755009\pi\)
−0.718147 + 0.695892i \(0.755009\pi\)
\(32\) −1.02358e8 −0.539257
\(33\) −1.67613e8 −0.745558
\(34\) 3.81878e6 0.0144142
\(35\) 2.73421e7 0.0879950
\(36\) −1.16630e8 −0.321474
\(37\) −5.53325e6 −0.0131181 −0.00655905 0.999978i \(-0.502088\pi\)
−0.00655905 + 0.999978i \(0.502088\pi\)
\(38\) −1.17934e7 −0.0241452
\(39\) −1.95745e8 −0.347405
\(40\) −1.55730e7 −0.0240459
\(41\) 6.36122e8 0.857490 0.428745 0.903426i \(-0.358956\pi\)
0.428745 + 0.903426i \(0.358956\pi\)
\(42\) −1.25066e8 −0.147662
\(43\) −1.32430e9 −1.37376 −0.686880 0.726771i \(-0.741020\pi\)
−0.686880 + 0.726771i \(0.741020\pi\)
\(44\) −1.36238e9 −1.24540
\(45\) −2.67772e7 −0.0216320
\(46\) 2.31116e8 0.165448
\(47\) 2.16637e9 1.37783 0.688913 0.724844i \(-0.258088\pi\)
0.688913 + 0.724844i \(0.258088\pi\)
\(48\) −9.11722e8 −0.516459
\(49\) 1.65813e9 0.838569
\(50\) 4.15040e8 0.187826
\(51\) 1.08712e8 0.0441207
\(52\) −1.59104e9 −0.580315
\(53\) 1.51378e8 0.0497215 0.0248607 0.999691i \(-0.492086\pi\)
0.0248607 + 0.999691i \(0.492086\pi\)
\(54\) 1.22482e8 0.0363000
\(55\) −3.12792e8 −0.0838031
\(56\) −2.07060e9 −0.502415
\(57\) −3.35733e8 −0.0739064
\(58\) 8.40244e8 0.168094
\(59\) −7.14924e8 −0.130189
\(60\) −2.17649e8 −0.0361347
\(61\) −1.17893e10 −1.78721 −0.893604 0.448856i \(-0.851832\pi\)
−0.893604 + 0.448856i \(0.851832\pi\)
\(62\) 1.95427e9 0.270914
\(63\) −3.56034e9 −0.451980
\(64\) −6.81026e9 −0.792818
\(65\) −3.65290e8 −0.0390494
\(66\) 1.43074e9 0.140627
\(67\) −1.16751e10 −1.05645 −0.528227 0.849103i \(-0.677143\pi\)
−0.528227 + 0.849103i \(0.677143\pi\)
\(68\) 8.83627e8 0.0737004
\(69\) 6.57937e9 0.506424
\(70\) −2.33392e8 −0.0165976
\(71\) 2.77195e10 1.82333 0.911665 0.410935i \(-0.134798\pi\)
0.911665 + 0.410935i \(0.134798\pi\)
\(72\) 2.02782e9 0.123510
\(73\) −9.19919e9 −0.519367 −0.259683 0.965694i \(-0.583618\pi\)
−0.259683 + 0.965694i \(0.583618\pi\)
\(74\) 4.72317e7 0.00247434
\(75\) 1.18153e10 0.574919
\(76\) −2.72888e9 −0.123455
\(77\) −4.15893e10 −1.75098
\(78\) 1.67088e9 0.0655275
\(79\) −2.72056e10 −0.994740 −0.497370 0.867538i \(-0.665701\pi\)
−0.497370 + 0.867538i \(0.665701\pi\)
\(80\) −1.70141e9 −0.0580516
\(81\) 3.48678e9 0.111111
\(82\) −5.42992e9 −0.161740
\(83\) 1.20624e10 0.336129 0.168064 0.985776i \(-0.446248\pi\)
0.168064 + 0.985776i \(0.446248\pi\)
\(84\) −2.89389e10 −0.755000
\(85\) 2.02873e8 0.00495930
\(86\) 1.13042e10 0.259119
\(87\) 2.39199e10 0.514521
\(88\) 2.36875e10 0.478481
\(89\) 5.23015e10 0.992818 0.496409 0.868089i \(-0.334652\pi\)
0.496409 + 0.868089i \(0.334652\pi\)
\(90\) 2.28570e8 0.00408023
\(91\) −4.85695e10 −0.815899
\(92\) 5.34780e10 0.845945
\(93\) 5.56338e10 0.829244
\(94\) −1.84921e10 −0.259886
\(95\) −6.26528e8 −0.00830732
\(96\) 2.48729e10 0.311340
\(97\) 1.76629e10 0.208842 0.104421 0.994533i \(-0.466701\pi\)
0.104421 + 0.994533i \(0.466701\pi\)
\(98\) −1.41537e10 −0.158171
\(99\) 4.07300e10 0.430448
\(100\) 9.60361e10 0.960361
\(101\) −1.32322e11 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(102\) −9.27964e8 −0.00832204
\(103\) 1.03204e11 0.877188 0.438594 0.898685i \(-0.355477\pi\)
0.438594 + 0.898685i \(0.355477\pi\)
\(104\) 2.76632e10 0.222956
\(105\) −6.64414e9 −0.0508039
\(106\) −1.29215e9 −0.00937847
\(107\) 1.16394e11 0.802270 0.401135 0.916019i \(-0.368616\pi\)
0.401135 + 0.916019i \(0.368616\pi\)
\(108\) 2.83411e10 0.185603
\(109\) −5.94860e10 −0.370313 −0.185156 0.982709i \(-0.559279\pi\)
−0.185156 + 0.982709i \(0.559279\pi\)
\(110\) 2.66998e9 0.0158070
\(111\) 1.34458e9 0.00757374
\(112\) −2.26222e11 −1.21293
\(113\) −5.72693e10 −0.292409 −0.146204 0.989254i \(-0.546706\pi\)
−0.146204 + 0.989254i \(0.546706\pi\)
\(114\) 2.86581e9 0.0139402
\(115\) 1.22781e10 0.0569237
\(116\) 1.94424e11 0.859470
\(117\) 4.75661e10 0.200574
\(118\) 6.10257e9 0.0245563
\(119\) 2.69743e10 0.103620
\(120\) 3.78423e9 0.0138829
\(121\) 1.90466e11 0.667571
\(122\) 1.00633e11 0.337104
\(123\) −1.54578e11 −0.495072
\(124\) 4.52199e11 1.38519
\(125\) 4.41914e10 0.129519
\(126\) 3.03910e10 0.0852524
\(127\) −5.52439e11 −1.48376 −0.741880 0.670533i \(-0.766066\pi\)
−0.741880 + 0.670533i \(0.766066\pi\)
\(128\) 2.67761e11 0.688798
\(129\) 3.21805e11 0.793140
\(130\) 3.11811e9 0.00736550
\(131\) −2.73769e11 −0.620000 −0.310000 0.950737i \(-0.600329\pi\)
−0.310000 + 0.950737i \(0.600329\pi\)
\(132\) 3.31059e11 0.719033
\(133\) −8.33042e10 −0.173573
\(134\) 9.96587e10 0.199269
\(135\) 6.50687e9 0.0124892
\(136\) −1.53635e10 −0.0283156
\(137\) −7.40534e11 −1.31094 −0.655469 0.755222i \(-0.727529\pi\)
−0.655469 + 0.755222i \(0.727529\pi\)
\(138\) −5.61613e10 −0.0955217
\(139\) −9.74103e11 −1.59229 −0.796147 0.605103i \(-0.793132\pi\)
−0.796147 + 0.605103i \(0.793132\pi\)
\(140\) −5.40045e10 −0.0848644
\(141\) −5.26428e11 −0.795488
\(142\) −2.36613e11 −0.343917
\(143\) 5.55631e11 0.777031
\(144\) 2.21549e11 0.298178
\(145\) 4.46381e10 0.0578338
\(146\) 7.85241e10 0.0979630
\(147\) −4.02924e11 −0.484148
\(148\) 1.09289e10 0.0126514
\(149\) 5.22558e11 0.582922 0.291461 0.956583i \(-0.405859\pi\)
0.291461 + 0.956583i \(0.405859\pi\)
\(150\) −1.00855e11 −0.108441
\(151\) 1.62306e12 1.68253 0.841264 0.540625i \(-0.181812\pi\)
0.841264 + 0.540625i \(0.181812\pi\)
\(152\) 4.74466e10 0.0474314
\(153\) −2.64170e10 −0.0254731
\(154\) 3.55005e11 0.330271
\(155\) 1.03821e11 0.0932097
\(156\) 3.86624e11 0.335045
\(157\) 1.27650e12 1.06801 0.534003 0.845483i \(-0.320687\pi\)
0.534003 + 0.845483i \(0.320687\pi\)
\(158\) 2.32226e11 0.187628
\(159\) −3.67847e10 −0.0287067
\(160\) 4.64166e10 0.0349956
\(161\) 1.63251e12 1.18936
\(162\) −2.97631e10 −0.0209578
\(163\) −1.34152e11 −0.0913196 −0.0456598 0.998957i \(-0.514539\pi\)
−0.0456598 + 0.998957i \(0.514539\pi\)
\(164\) −1.25643e12 −0.826983
\(165\) 7.60083e10 0.0483838
\(166\) −1.02965e11 −0.0634007
\(167\) 2.30164e12 1.37118 0.685592 0.727986i \(-0.259543\pi\)
0.685592 + 0.727986i \(0.259543\pi\)
\(168\) 5.03157e11 0.290070
\(169\) −1.14327e12 −0.637930
\(170\) −1.73172e9 −0.000935424 0
\(171\) 8.15831e10 0.0426699
\(172\) 2.61568e12 1.32488
\(173\) −5.04643e11 −0.247589 −0.123794 0.992308i \(-0.539506\pi\)
−0.123794 + 0.992308i \(0.539506\pi\)
\(174\) −2.04179e11 −0.0970490
\(175\) 2.93168e12 1.35023
\(176\) 2.58796e12 1.15515
\(177\) 1.73727e11 0.0751646
\(178\) −4.46445e11 −0.187265
\(179\) 2.45301e12 0.997718 0.498859 0.866683i \(-0.333753\pi\)
0.498859 + 0.866683i \(0.333753\pi\)
\(180\) 5.28887e10 0.0208624
\(181\) 7.78232e11 0.297767 0.148884 0.988855i \(-0.452432\pi\)
0.148884 + 0.988855i \(0.452432\pi\)
\(182\) 4.14588e11 0.153895
\(183\) 2.86481e12 1.03185
\(184\) −9.29813e11 −0.325011
\(185\) 2.50919e9 0.000851313 0
\(186\) −4.74889e11 −0.156412
\(187\) −3.08584e11 −0.0986835
\(188\) −4.27888e12 −1.32881
\(189\) 8.65163e11 0.260950
\(190\) 5.34803e9 0.00156693
\(191\) −3.35775e11 −0.0955795 −0.0477898 0.998857i \(-0.515218\pi\)
−0.0477898 + 0.998857i \(0.515218\pi\)
\(192\) 1.65489e12 0.457734
\(193\) 2.69775e12 0.725164 0.362582 0.931952i \(-0.381895\pi\)
0.362582 + 0.931952i \(0.381895\pi\)
\(194\) −1.50770e11 −0.0393918
\(195\) 8.87655e10 0.0225452
\(196\) −3.27503e12 −0.808735
\(197\) 1.22410e12 0.293936 0.146968 0.989141i \(-0.453049\pi\)
0.146968 + 0.989141i \(0.453049\pi\)
\(198\) −3.47670e11 −0.0811912
\(199\) −4.55968e12 −1.03572 −0.517860 0.855465i \(-0.673271\pi\)
−0.517860 + 0.855465i \(0.673271\pi\)
\(200\) −1.66976e12 −0.368969
\(201\) 2.83706e12 0.609944
\(202\) 1.12950e12 0.236295
\(203\) 5.93515e12 1.20838
\(204\) −2.14721e11 −0.0425509
\(205\) −2.88465e11 −0.0556477
\(206\) −8.80948e11 −0.165455
\(207\) −1.59879e12 −0.292384
\(208\) 3.02232e12 0.538261
\(209\) 9.52992e11 0.165305
\(210\) 5.67142e10 0.00958265
\(211\) 4.29393e12 0.706807 0.353404 0.935471i \(-0.385024\pi\)
0.353404 + 0.935471i \(0.385024\pi\)
\(212\) −2.98991e11 −0.0479525
\(213\) −6.73585e12 −1.05270
\(214\) −9.93538e11 −0.151324
\(215\) 6.00538e11 0.0891515
\(216\) −4.92761e11 −0.0713085
\(217\) 1.38042e13 1.94753
\(218\) 5.07771e11 0.0698485
\(219\) 2.23540e12 0.299856
\(220\) 6.17806e11 0.0808216
\(221\) −3.60376e11 −0.0459832
\(222\) −1.14773e10 −0.00142856
\(223\) −4.18410e12 −0.508072 −0.254036 0.967195i \(-0.581758\pi\)
−0.254036 + 0.967195i \(0.581758\pi\)
\(224\) 6.17163e12 0.731199
\(225\) −2.87111e12 −0.331930
\(226\) 4.88849e11 0.0551542
\(227\) −8.36479e12 −0.921113 −0.460557 0.887630i \(-0.652350\pi\)
−0.460557 + 0.887630i \(0.652350\pi\)
\(228\) 6.63118e11 0.0712770
\(229\) −6.49618e12 −0.681652 −0.340826 0.940126i \(-0.610707\pi\)
−0.340826 + 0.940126i \(0.610707\pi\)
\(230\) −1.04805e11 −0.0107369
\(231\) 1.01062e13 1.01093
\(232\) −3.38042e12 −0.330207
\(233\) 1.54565e13 1.47453 0.737264 0.675604i \(-0.236117\pi\)
0.737264 + 0.675604i \(0.236117\pi\)
\(234\) −4.06023e11 −0.0378323
\(235\) −9.82394e11 −0.0894154
\(236\) 1.41207e12 0.125557
\(237\) 6.61097e12 0.574314
\(238\) −2.30252e11 −0.0195448
\(239\) 1.54290e13 1.27982 0.639912 0.768448i \(-0.278971\pi\)
0.639912 + 0.768448i \(0.278971\pi\)
\(240\) 4.13443e11 0.0335161
\(241\) −2.65790e11 −0.0210594 −0.0105297 0.999945i \(-0.503352\pi\)
−0.0105297 + 0.999945i \(0.503352\pi\)
\(242\) −1.62581e12 −0.125917
\(243\) −8.47289e11 −0.0641500
\(244\) 2.32856e13 1.72362
\(245\) −7.51918e11 −0.0544198
\(246\) 1.31947e12 0.0933806
\(247\) 1.11294e12 0.0770263
\(248\) −7.86231e12 −0.532189
\(249\) −2.93117e12 −0.194064
\(250\) −3.77216e11 −0.0244298
\(251\) 4.95418e12 0.313882 0.156941 0.987608i \(-0.449837\pi\)
0.156941 + 0.987608i \(0.449837\pi\)
\(252\) 7.03216e12 0.435899
\(253\) −1.86758e13 −1.13271
\(254\) 4.71560e12 0.279867
\(255\) −4.92982e10 −0.00286325
\(256\) 1.16618e13 0.662897
\(257\) 3.07033e12 0.170826 0.0854129 0.996346i \(-0.472779\pi\)
0.0854129 + 0.996346i \(0.472779\pi\)
\(258\) −2.74692e12 −0.149602
\(259\) 3.33626e11 0.0177873
\(260\) 7.21498e11 0.0376601
\(261\) −5.81253e12 −0.297059
\(262\) 2.33688e12 0.116944
\(263\) 2.00792e13 0.983989 0.491995 0.870598i \(-0.336268\pi\)
0.491995 + 0.870598i \(0.336268\pi\)
\(264\) −5.75607e12 −0.276251
\(265\) −6.86459e10 −0.00322673
\(266\) 7.11082e11 0.0327394
\(267\) −1.27093e13 −0.573204
\(268\) 2.30600e13 1.01887
\(269\) −2.48330e13 −1.07496 −0.537478 0.843277i \(-0.680623\pi\)
−0.537478 + 0.843277i \(0.680623\pi\)
\(270\) −5.55424e10 −0.00235572
\(271\) −6.94257e12 −0.288529 −0.144264 0.989539i \(-0.546082\pi\)
−0.144264 + 0.989539i \(0.546082\pi\)
\(272\) −1.67853e12 −0.0683595
\(273\) 1.18024e13 0.471059
\(274\) 6.32118e12 0.247269
\(275\) −3.35381e13 −1.28591
\(276\) −1.29952e13 −0.488406
\(277\) 4.19556e13 1.54579 0.772897 0.634532i \(-0.218807\pi\)
0.772897 + 0.634532i \(0.218807\pi\)
\(278\) 8.31491e12 0.300339
\(279\) −1.35190e13 −0.478764
\(280\) 9.38967e11 0.0326048
\(281\) −2.80186e13 −0.954030 −0.477015 0.878895i \(-0.658281\pi\)
−0.477015 + 0.878895i \(0.658281\pi\)
\(282\) 4.49357e12 0.150045
\(283\) 1.35272e13 0.442979 0.221490 0.975163i \(-0.428908\pi\)
0.221490 + 0.975163i \(0.428908\pi\)
\(284\) −5.47499e13 −1.75846
\(285\) 1.52246e11 0.00479623
\(286\) −4.74285e12 −0.146564
\(287\) −3.83548e13 −1.16270
\(288\) −6.04412e12 −0.179752
\(289\) −3.40718e13 −0.994160
\(290\) −3.81029e11 −0.0109086
\(291\) −4.29209e12 −0.120575
\(292\) 1.81697e13 0.500889
\(293\) 4.64179e13 1.25578 0.627890 0.778302i \(-0.283919\pi\)
0.627890 + 0.778302i \(0.283919\pi\)
\(294\) 3.43935e12 0.0913201
\(295\) 3.24200e11 0.00844874
\(296\) −1.90020e11 −0.00486065
\(297\) −9.89739e12 −0.248519
\(298\) −4.46054e12 −0.109951
\(299\) −2.18103e13 −0.527802
\(300\) −2.33368e13 −0.554465
\(301\) 7.98484e13 1.86273
\(302\) −1.38544e13 −0.317359
\(303\) 3.21544e13 0.723278
\(304\) 5.18375e12 0.114509
\(305\) 5.34617e12 0.115983
\(306\) 2.25495e11 0.00480473
\(307\) −5.60175e13 −1.17236 −0.586182 0.810179i \(-0.699370\pi\)
−0.586182 + 0.810179i \(0.699370\pi\)
\(308\) 8.21445e13 1.68869
\(309\) −2.50786e13 −0.506444
\(310\) −8.86214e11 −0.0175812
\(311\) −5.49934e13 −1.07184 −0.535918 0.844270i \(-0.680035\pi\)
−0.535918 + 0.844270i \(0.680035\pi\)
\(312\) −6.72216e12 −0.128724
\(313\) 3.63968e13 0.684808 0.342404 0.939553i \(-0.388759\pi\)
0.342404 + 0.939553i \(0.388759\pi\)
\(314\) −1.08962e13 −0.201447
\(315\) 1.61453e12 0.0293317
\(316\) 5.37348e13 0.959350
\(317\) 9.89968e12 0.173698 0.0868491 0.996221i \(-0.472320\pi\)
0.0868491 + 0.996221i \(0.472320\pi\)
\(318\) 3.13994e11 0.00541466
\(319\) −6.78976e13 −1.15082
\(320\) 3.08828e12 0.0514508
\(321\) −2.82838e13 −0.463191
\(322\) −1.39351e13 −0.224338
\(323\) −6.18101e11 −0.00978240
\(324\) −6.88688e12 −0.107158
\(325\) −3.91671e13 −0.599188
\(326\) 1.14511e12 0.0172247
\(327\) 1.44551e13 0.213800
\(328\) 2.18453e13 0.317725
\(329\) −1.30621e14 −1.86825
\(330\) −6.48805e11 −0.00912615
\(331\) 7.57286e13 1.04763 0.523813 0.851833i \(-0.324509\pi\)
0.523813 + 0.851833i \(0.324509\pi\)
\(332\) −2.38250e13 −0.324170
\(333\) −3.26733e11 −0.00437270
\(334\) −1.96467e13 −0.258633
\(335\) 5.29438e12 0.0685597
\(336\) 5.49720e13 0.700287
\(337\) 5.00784e13 0.627605 0.313802 0.949488i \(-0.398397\pi\)
0.313802 + 0.949488i \(0.398397\pi\)
\(338\) 9.75895e12 0.120326
\(339\) 1.39164e13 0.168822
\(340\) −4.00702e11 −0.00478286
\(341\) −1.57919e14 −1.85475
\(342\) −6.96391e11 −0.00804840
\(343\) 1.92461e13 0.218890
\(344\) −4.54784e13 −0.509019
\(345\) −2.98358e12 −0.0328649
\(346\) 4.30762e12 0.0467002
\(347\) 1.10923e14 1.18361 0.591805 0.806081i \(-0.298416\pi\)
0.591805 + 0.806081i \(0.298416\pi\)
\(348\) −4.72450e13 −0.496215
\(349\) −1.00342e14 −1.03739 −0.518696 0.854959i \(-0.673582\pi\)
−0.518696 + 0.854959i \(0.673582\pi\)
\(350\) −2.50247e13 −0.254680
\(351\) −1.15586e13 −0.115802
\(352\) −7.06029e13 −0.696366
\(353\) −4.16653e13 −0.404589 −0.202295 0.979325i \(-0.564840\pi\)
−0.202295 + 0.979325i \(0.564840\pi\)
\(354\) −1.48293e12 −0.0141776
\(355\) −1.25701e13 −0.118327
\(356\) −1.03303e14 −0.957496
\(357\) −6.55477e12 −0.0598249
\(358\) −2.09388e13 −0.188190
\(359\) 6.01394e13 0.532279 0.266139 0.963935i \(-0.414252\pi\)
0.266139 + 0.963935i \(0.414252\pi\)
\(360\) −9.19567e11 −0.00801530
\(361\) −1.14581e14 −0.983614
\(362\) −6.64297e12 −0.0561649
\(363\) −4.62832e13 −0.385422
\(364\) 9.59315e13 0.786871
\(365\) 4.17160e12 0.0337048
\(366\) −2.44539e13 −0.194627
\(367\) 1.65938e13 0.130102 0.0650508 0.997882i \(-0.479279\pi\)
0.0650508 + 0.997882i \(0.479279\pi\)
\(368\) −1.01586e14 −0.784641
\(369\) 3.75624e13 0.285830
\(370\) −2.14184e10 −0.000160575 0
\(371\) −9.12726e12 −0.0674193
\(372\) −1.09884e14 −0.799742
\(373\) −2.44863e14 −1.75600 −0.878000 0.478660i \(-0.841123\pi\)
−0.878000 + 0.478660i \(0.841123\pi\)
\(374\) 2.63407e12 0.0186137
\(375\) −1.07385e13 −0.0747777
\(376\) 7.43961e13 0.510525
\(377\) −7.92934e13 −0.536241
\(378\) −7.38501e12 −0.0492205
\(379\) 6.92794e12 0.0455081 0.0227540 0.999741i \(-0.492757\pi\)
0.0227540 + 0.999741i \(0.492757\pi\)
\(380\) 1.23748e12 0.00801176
\(381\) 1.34243e14 0.856649
\(382\) 2.86617e12 0.0180282
\(383\) 2.07697e13 0.128777 0.0643883 0.997925i \(-0.479490\pi\)
0.0643883 + 0.997925i \(0.479490\pi\)
\(384\) −6.50658e13 −0.397678
\(385\) 1.88597e13 0.113632
\(386\) −2.30279e13 −0.136781
\(387\) −7.81987e13 −0.457920
\(388\) −3.48867e13 −0.201412
\(389\) 2.45599e14 1.39799 0.698996 0.715126i \(-0.253631\pi\)
0.698996 + 0.715126i \(0.253631\pi\)
\(390\) −7.57700e11 −0.00425248
\(391\) 1.21129e13 0.0670312
\(392\) 5.69423e13 0.310715
\(393\) 6.65258e13 0.357957
\(394\) −1.04489e13 −0.0554423
\(395\) 1.23371e13 0.0645547
\(396\) −8.04474e13 −0.415134
\(397\) 1.42624e14 0.725846 0.362923 0.931819i \(-0.381779\pi\)
0.362923 + 0.931819i \(0.381779\pi\)
\(398\) 3.89213e13 0.195358
\(399\) 2.02429e13 0.100213
\(400\) −1.82429e14 −0.890766
\(401\) 1.19444e14 0.575268 0.287634 0.957740i \(-0.407131\pi\)
0.287634 + 0.957740i \(0.407131\pi\)
\(402\) −2.42171e13 −0.115048
\(403\) −1.84424e14 −0.864250
\(404\) 2.61355e14 1.20818
\(405\) −1.58117e12 −0.00721067
\(406\) −5.06623e13 −0.227925
\(407\) −3.81665e12 −0.0169400
\(408\) 3.73333e12 0.0163480
\(409\) −4.27237e14 −1.84583 −0.922913 0.385008i \(-0.874199\pi\)
−0.922913 + 0.385008i \(0.874199\pi\)
\(410\) 2.46233e12 0.0104963
\(411\) 1.79950e14 0.756870
\(412\) −2.03842e14 −0.845979
\(413\) 4.31062e13 0.176528
\(414\) 1.36472e13 0.0551495
\(415\) −5.47001e12 −0.0218134
\(416\) −8.24527e13 −0.324483
\(417\) 2.36707e14 0.919311
\(418\) −8.13472e12 −0.0311798
\(419\) 2.61164e14 0.987953 0.493977 0.869475i \(-0.335543\pi\)
0.493977 + 0.869475i \(0.335543\pi\)
\(420\) 1.31231e13 0.0489965
\(421\) 1.06415e14 0.392150 0.196075 0.980589i \(-0.437180\pi\)
0.196075 + 0.980589i \(0.437180\pi\)
\(422\) −3.66528e13 −0.133318
\(423\) 1.27922e14 0.459275
\(424\) 5.19851e12 0.0184233
\(425\) 2.17525e13 0.0760974
\(426\) 5.74970e13 0.198560
\(427\) 7.10835e14 2.42334
\(428\) −2.29895e14 −0.773727
\(429\) −1.35018e14 −0.448619
\(430\) −5.12617e12 −0.0168158
\(431\) −4.12476e14 −1.33590 −0.667950 0.744206i \(-0.732828\pi\)
−0.667950 + 0.744206i \(0.732828\pi\)
\(432\) −5.38363e13 −0.172153
\(433\) −4.09476e14 −1.29284 −0.646421 0.762981i \(-0.723735\pi\)
−0.646421 + 0.762981i \(0.723735\pi\)
\(434\) −1.17832e14 −0.367343
\(435\) −1.08471e13 −0.0333903
\(436\) 1.17493e14 0.357138
\(437\) −3.74081e13 −0.112284
\(438\) −1.90813e13 −0.0565590
\(439\) 2.55895e14 0.749042 0.374521 0.927218i \(-0.377807\pi\)
0.374521 + 0.927218i \(0.377807\pi\)
\(440\) −1.07417e13 −0.0310515
\(441\) 9.79107e13 0.279523
\(442\) 3.07616e12 0.00867335
\(443\) −1.36522e14 −0.380175 −0.190087 0.981767i \(-0.560877\pi\)
−0.190087 + 0.981767i \(0.560877\pi\)
\(444\) −2.65573e12 −0.00730429
\(445\) −2.37174e13 −0.0644299
\(446\) 3.57153e13 0.0958326
\(447\) −1.26982e14 −0.336550
\(448\) 4.10623e14 1.07501
\(449\) −3.84323e14 −0.993899 −0.496949 0.867780i \(-0.665546\pi\)
−0.496949 + 0.867780i \(0.665546\pi\)
\(450\) 2.45077e13 0.0626086
\(451\) 4.38775e14 1.10732
\(452\) 1.13115e14 0.282006
\(453\) −3.94404e14 −0.971408
\(454\) 7.14017e13 0.173741
\(455\) 2.20251e13 0.0529486
\(456\) −1.15295e13 −0.0273845
\(457\) 2.79727e14 0.656440 0.328220 0.944601i \(-0.393551\pi\)
0.328220 + 0.944601i \(0.393551\pi\)
\(458\) 5.54512e13 0.128573
\(459\) 6.41934e12 0.0147069
\(460\) −2.42509e13 −0.0548984
\(461\) 7.92191e14 1.77205 0.886023 0.463642i \(-0.153458\pi\)
0.886023 + 0.463642i \(0.153458\pi\)
\(462\) −8.62662e13 −0.190682
\(463\) 6.73166e14 1.47037 0.735185 0.677866i \(-0.237095\pi\)
0.735185 + 0.677866i \(0.237095\pi\)
\(464\) −3.69325e14 −0.797186
\(465\) −2.52285e13 −0.0538147
\(466\) −1.31936e14 −0.278126
\(467\) 2.40608e14 0.501266 0.250633 0.968082i \(-0.419361\pi\)
0.250633 + 0.968082i \(0.419361\pi\)
\(468\) −9.39495e13 −0.193438
\(469\) 7.03949e14 1.43249
\(470\) 8.38569e12 0.0168655
\(471\) −3.10190e14 −0.616613
\(472\) −2.45515e13 −0.0482389
\(473\) −9.13459e14 −1.77400
\(474\) −5.64310e13 −0.108327
\(475\) −6.71776e13 −0.127471
\(476\) −5.32780e13 −0.0999332
\(477\) 8.93869e12 0.0165738
\(478\) −1.31702e14 −0.241401
\(479\) −9.11031e14 −1.65078 −0.825388 0.564566i \(-0.809043\pi\)
−0.825388 + 0.564566i \(0.809043\pi\)
\(480\) −1.12792e13 −0.0202047
\(481\) −4.45723e12 −0.00789346
\(482\) 2.26878e12 0.00397222
\(483\) −3.96701e14 −0.686680
\(484\) −3.76196e14 −0.643821
\(485\) −8.00968e12 −0.0135530
\(486\) 7.23243e12 0.0121000
\(487\) 3.90079e14 0.645273 0.322637 0.946523i \(-0.395431\pi\)
0.322637 + 0.946523i \(0.395431\pi\)
\(488\) −4.04862e14 −0.662214
\(489\) 3.25988e13 0.0527234
\(490\) 6.41835e12 0.0102647
\(491\) 5.29891e14 0.837989 0.418994 0.907989i \(-0.362383\pi\)
0.418994 + 0.907989i \(0.362383\pi\)
\(492\) 3.05312e14 0.477459
\(493\) 4.40376e13 0.0681030
\(494\) −9.50004e12 −0.0145287
\(495\) −1.84700e13 −0.0279344
\(496\) −8.58991e14 −1.28481
\(497\) −1.67134e15 −2.47232
\(498\) 2.50204e13 0.0366044
\(499\) 5.03448e13 0.0728453 0.0364227 0.999336i \(-0.488404\pi\)
0.0364227 + 0.999336i \(0.488404\pi\)
\(500\) −8.72841e13 −0.124911
\(501\) −5.59298e14 −0.791654
\(502\) −4.22888e13 −0.0592045
\(503\) 8.51006e14 1.17844 0.589221 0.807972i \(-0.299435\pi\)
0.589221 + 0.807972i \(0.299435\pi\)
\(504\) −1.22267e14 −0.167472
\(505\) 6.00049e13 0.0812988
\(506\) 1.59416e14 0.213651
\(507\) 2.77815e14 0.368309
\(508\) 1.09114e15 1.43097
\(509\) −4.39929e13 −0.0570736 −0.0285368 0.999593i \(-0.509085\pi\)
−0.0285368 + 0.999593i \(0.509085\pi\)
\(510\) 4.20808e11 0.000540068 0
\(511\) 5.54663e14 0.704229
\(512\) −6.47919e14 −0.813834
\(513\) −1.98247e13 −0.0246355
\(514\) −2.62083e13 −0.0322212
\(515\) −4.68005e13 −0.0569260
\(516\) −6.35610e14 −0.764922
\(517\) 1.49429e15 1.77925
\(518\) −2.84782e12 −0.00335505
\(519\) 1.22628e14 0.142945
\(520\) −1.25446e13 −0.0144690
\(521\) 1.63088e15 1.86130 0.930649 0.365914i \(-0.119244\pi\)
0.930649 + 0.365914i \(0.119244\pi\)
\(522\) 4.96156e13 0.0560312
\(523\) 7.82493e14 0.874422 0.437211 0.899359i \(-0.355966\pi\)
0.437211 + 0.899359i \(0.355966\pi\)
\(524\) 5.40731e14 0.597942
\(525\) −7.12398e14 −0.779555
\(526\) −1.71396e14 −0.185600
\(527\) 1.02425e14 0.109760
\(528\) −6.28875e14 −0.666926
\(529\) −2.19723e14 −0.230605
\(530\) 5.85959e11 0.000608626 0
\(531\) −4.22156e13 −0.0433963
\(532\) 1.64537e14 0.167398
\(533\) 5.12419e14 0.515971
\(534\) 1.08486e14 0.108118
\(535\) −5.27818e13 −0.0520641
\(536\) −4.00941e14 −0.391448
\(537\) −5.96082e14 −0.576033
\(538\) 2.11974e14 0.202759
\(539\) 1.14372e15 1.08288
\(540\) −1.28520e13 −0.0120449
\(541\) 1.46668e15 1.36066 0.680331 0.732905i \(-0.261836\pi\)
0.680331 + 0.732905i \(0.261836\pi\)
\(542\) 5.92616e13 0.0544223
\(543\) −1.89110e14 −0.171916
\(544\) 4.57923e13 0.0412096
\(545\) 2.69754e13 0.0240318
\(546\) −1.00745e14 −0.0888513
\(547\) 2.01595e15 1.76015 0.880074 0.474836i \(-0.157493\pi\)
0.880074 + 0.474836i \(0.157493\pi\)
\(548\) 1.46266e15 1.26430
\(549\) −6.96148e14 −0.595736
\(550\) 2.86281e14 0.242548
\(551\) −1.36000e14 −0.114079
\(552\) 2.25945e14 0.187645
\(553\) 1.64036e15 1.34881
\(554\) −3.58132e14 −0.291568
\(555\) −6.09733e11 −0.000491506 0
\(556\) 1.92399e15 1.53564
\(557\) 7.11658e14 0.562429 0.281215 0.959645i \(-0.409263\pi\)
0.281215 + 0.959645i \(0.409263\pi\)
\(558\) 1.15398e14 0.0903046
\(559\) −1.06677e15 −0.826622
\(560\) 1.02586e14 0.0787144
\(561\) 7.49859e13 0.0569750
\(562\) 2.39166e14 0.179949
\(563\) −8.54038e14 −0.636328 −0.318164 0.948036i \(-0.603066\pi\)
−0.318164 + 0.948036i \(0.603066\pi\)
\(564\) 1.03977e15 0.767186
\(565\) 2.59702e13 0.0189762
\(566\) −1.15468e14 −0.0835548
\(567\) −2.10235e14 −0.150660
\(568\) 9.51928e14 0.675598
\(569\) 2.52874e14 0.177741 0.0888704 0.996043i \(-0.471674\pi\)
0.0888704 + 0.996043i \(0.471674\pi\)
\(570\) −1.29957e12 −0.000904666 0
\(571\) 1.63683e15 1.12851 0.564253 0.825602i \(-0.309164\pi\)
0.564253 + 0.825602i \(0.309164\pi\)
\(572\) −1.09745e15 −0.749386
\(573\) 8.15933e13 0.0551829
\(574\) 3.27395e14 0.219309
\(575\) 1.31648e15 0.873458
\(576\) −4.02139e14 −0.264273
\(577\) −6.96578e13 −0.0453422 −0.0226711 0.999743i \(-0.507217\pi\)
−0.0226711 + 0.999743i \(0.507217\pi\)
\(578\) 2.90836e14 0.187519
\(579\) −6.55553e14 −0.418674
\(580\) −8.81663e13 −0.0557762
\(581\) −7.27302e14 −0.455770
\(582\) 3.66371e13 0.0227429
\(583\) 1.04415e14 0.0642076
\(584\) −3.15913e14 −0.192441
\(585\) −2.15700e13 −0.0130165
\(586\) −3.96222e14 −0.236865
\(587\) 2.43685e15 1.44318 0.721589 0.692322i \(-0.243412\pi\)
0.721589 + 0.692322i \(0.243412\pi\)
\(588\) 7.95831e14 0.466923
\(589\) −3.16315e14 −0.183859
\(590\) −2.76736e12 −0.00159360
\(591\) −2.97456e14 −0.169704
\(592\) −2.07605e13 −0.0117346
\(593\) −8.54193e14 −0.478360 −0.239180 0.970975i \(-0.576879\pi\)
−0.239180 + 0.970975i \(0.576879\pi\)
\(594\) 8.44839e13 0.0468758
\(595\) −1.22322e13 −0.00672451
\(596\) −1.03212e15 −0.562183
\(597\) 1.10800e15 0.597973
\(598\) 1.86173e14 0.0995541
\(599\) 1.88150e15 0.996910 0.498455 0.866916i \(-0.333901\pi\)
0.498455 + 0.866916i \(0.333901\pi\)
\(600\) 4.05753e14 0.213024
\(601\) 1.44918e15 0.753899 0.376949 0.926234i \(-0.376973\pi\)
0.376949 + 0.926234i \(0.376973\pi\)
\(602\) −6.81584e14 −0.351349
\(603\) −6.89405e14 −0.352151
\(604\) −3.20577e15 −1.62267
\(605\) −8.63714e13 −0.0433227
\(606\) −2.74469e14 −0.136425
\(607\) 7.45733e14 0.367321 0.183660 0.982990i \(-0.441205\pi\)
0.183660 + 0.982990i \(0.441205\pi\)
\(608\) −1.41419e14 −0.0690300
\(609\) −1.44224e15 −0.697658
\(610\) −4.56347e13 −0.0218767
\(611\) 1.74509e15 0.829069
\(612\) 5.21773e13 0.0245668
\(613\) 7.84364e14 0.366004 0.183002 0.983113i \(-0.441419\pi\)
0.183002 + 0.983113i \(0.441419\pi\)
\(614\) 4.78164e14 0.221131
\(615\) 7.00970e13 0.0321282
\(616\) −1.42823e15 −0.648791
\(617\) −2.89902e15 −1.30522 −0.652609 0.757695i \(-0.726326\pi\)
−0.652609 + 0.757695i \(0.726326\pi\)
\(618\) 2.14070e14 0.0955256
\(619\) −1.36922e15 −0.605584 −0.302792 0.953057i \(-0.597919\pi\)
−0.302792 + 0.953057i \(0.597919\pi\)
\(620\) −2.05061e14 −0.0898935
\(621\) 3.88505e14 0.168808
\(622\) 4.69422e14 0.202170
\(623\) −3.15351e15 −1.34620
\(624\) −7.34425e14 −0.310765
\(625\) 2.35411e15 0.987383
\(626\) −3.10682e14 −0.129169
\(627\) −2.31577e14 −0.0954386
\(628\) −2.52127e15 −1.03001
\(629\) 2.47544e12 0.00100248
\(630\) −1.37815e13 −0.00553254
\(631\) 4.34726e14 0.173003 0.0865017 0.996252i \(-0.472431\pi\)
0.0865017 + 0.996252i \(0.472431\pi\)
\(632\) −9.34279e14 −0.368581
\(633\) −1.04342e15 −0.408075
\(634\) −8.45034e13 −0.0327630
\(635\) 2.50517e14 0.0962901
\(636\) 7.26549e13 0.0276854
\(637\) 1.33568e15 0.504586
\(638\) 5.79572e14 0.217067
\(639\) 1.63681e15 0.607777
\(640\) −1.21423e14 −0.0447003
\(641\) −3.46314e15 −1.26401 −0.632005 0.774965i \(-0.717768\pi\)
−0.632005 + 0.774965i \(0.717768\pi\)
\(642\) 2.41430e14 0.0873671
\(643\) 2.32821e15 0.835338 0.417669 0.908599i \(-0.362847\pi\)
0.417669 + 0.908599i \(0.362847\pi\)
\(644\) −3.22444e15 −1.14705
\(645\) −1.45931e14 −0.0514716
\(646\) 5.27609e12 0.00184516
\(647\) 2.36081e15 0.818631 0.409316 0.912393i \(-0.365767\pi\)
0.409316 + 0.912393i \(0.365767\pi\)
\(648\) 1.19741e14 0.0411700
\(649\) −4.93131e14 −0.168119
\(650\) 3.34330e14 0.113019
\(651\) −3.35442e15 −1.12440
\(652\) 2.64968e14 0.0880707
\(653\) −3.74274e15 −1.23358 −0.616789 0.787129i \(-0.711567\pi\)
−0.616789 + 0.787129i \(0.711567\pi\)
\(654\) −1.23388e14 −0.0403270
\(655\) 1.24147e14 0.0402355
\(656\) 2.38669e15 0.767053
\(657\) −5.43203e14 −0.173122
\(658\) 1.11497e15 0.352389
\(659\) −1.80088e15 −0.564436 −0.282218 0.959350i \(-0.591070\pi\)
−0.282218 + 0.959350i \(0.591070\pi\)
\(660\) −1.50127e14 −0.0466624
\(661\) 3.82669e14 0.117955 0.0589774 0.998259i \(-0.481216\pi\)
0.0589774 + 0.998259i \(0.481216\pi\)
\(662\) −6.46418e14 −0.197603
\(663\) 8.75715e13 0.0265484
\(664\) 4.14241e14 0.124546
\(665\) 3.77763e13 0.0112642
\(666\) 2.78899e12 0.000824780 0
\(667\) 2.66520e15 0.781696
\(668\) −4.54605e15 −1.32240
\(669\) 1.01674e15 0.293335
\(670\) −4.51927e13 −0.0129317
\(671\) −8.13189e15 −2.30790
\(672\) −1.49971e15 −0.422158
\(673\) 4.10989e15 1.14749 0.573743 0.819035i \(-0.305491\pi\)
0.573743 + 0.819035i \(0.305491\pi\)
\(674\) −4.27468e14 −0.118379
\(675\) 6.97680e14 0.191640
\(676\) 2.25812e15 0.615234
\(677\) 5.24818e15 1.41831 0.709155 0.705052i \(-0.249076\pi\)
0.709155 + 0.705052i \(0.249076\pi\)
\(678\) −1.18790e14 −0.0318433
\(679\) −1.06498e15 −0.283177
\(680\) 6.96695e12 0.00183757
\(681\) 2.03264e15 0.531805
\(682\) 1.34799e15 0.349843
\(683\) 1.25331e15 0.322659 0.161329 0.986901i \(-0.448422\pi\)
0.161329 + 0.986901i \(0.448422\pi\)
\(684\) −1.61138e14 −0.0411518
\(685\) 3.35814e14 0.0850746
\(686\) −1.64285e14 −0.0412871
\(687\) 1.57857e15 0.393552
\(688\) −4.96871e15 −1.22887
\(689\) 1.21940e14 0.0299185
\(690\) 2.54677e13 0.00619898
\(691\) −6.34553e15 −1.53228 −0.766141 0.642672i \(-0.777826\pi\)
−0.766141 + 0.642672i \(0.777826\pi\)
\(692\) 9.96740e14 0.238780
\(693\) −2.45580e15 −0.583661
\(694\) −9.46833e14 −0.223253
\(695\) 4.41731e14 0.103334
\(696\) 8.21441e14 0.190645
\(697\) −2.84585e14 −0.0655287
\(698\) 8.56517e14 0.195673
\(699\) −3.75593e15 −0.851320
\(700\) −5.79047e15 −1.30219
\(701\) −5.53122e15 −1.23416 −0.617081 0.786900i \(-0.711685\pi\)
−0.617081 + 0.786900i \(0.711685\pi\)
\(702\) 9.86635e13 0.0218425
\(703\) −7.64483e12 −0.00167924
\(704\) −4.69749e15 −1.02380
\(705\) 2.38722e14 0.0516240
\(706\) 3.55654e14 0.0763136
\(707\) 7.97835e15 1.69866
\(708\) −3.43134e14 −0.0724904
\(709\) −7.56055e15 −1.58489 −0.792445 0.609943i \(-0.791192\pi\)
−0.792445 + 0.609943i \(0.791192\pi\)
\(710\) 1.07298e14 0.0223188
\(711\) −1.60646e15 −0.331580
\(712\) 1.79611e15 0.367868
\(713\) 6.19884e15 1.25985
\(714\) 5.59513e13 0.0112842
\(715\) −2.51965e14 −0.0504262
\(716\) −4.84503e15 −0.962222
\(717\) −3.74926e15 −0.738907
\(718\) −5.13348e14 −0.100399
\(719\) 3.52293e15 0.683747 0.341874 0.939746i \(-0.388939\pi\)
0.341874 + 0.939746i \(0.388939\pi\)
\(720\) −1.00467e14 −0.0193505
\(721\) −6.22266e15 −1.18941
\(722\) 9.78064e14 0.185529
\(723\) 6.45870e13 0.0121586
\(724\) −1.53712e15 −0.287174
\(725\) 4.78619e15 0.887423
\(726\) 3.95072e14 0.0726984
\(727\) −5.82118e15 −1.06309 −0.531547 0.847029i \(-0.678389\pi\)
−0.531547 + 0.847029i \(0.678389\pi\)
\(728\) −1.66795e15 −0.302315
\(729\) 2.05891e14 0.0370370
\(730\) −3.56087e13 −0.00635741
\(731\) 5.92460e14 0.104982
\(732\) −5.65839e15 −0.995135
\(733\) 9.33213e15 1.62895 0.814477 0.580195i \(-0.197024\pi\)
0.814477 + 0.580195i \(0.197024\pi\)
\(734\) −1.41644e14 −0.0245398
\(735\) 1.82716e14 0.0314193
\(736\) 2.77139e15 0.473010
\(737\) −8.05312e15 −1.36425
\(738\) −3.20631e14 −0.0539133
\(739\) −8.74651e14 −0.145979 −0.0729895 0.997333i \(-0.523254\pi\)
−0.0729895 + 0.997333i \(0.523254\pi\)
\(740\) −4.95600e12 −0.000821025 0
\(741\) −2.70445e14 −0.0444712
\(742\) 7.79101e13 0.0127166
\(743\) −1.20583e16 −1.95365 −0.976824 0.214045i \(-0.931336\pi\)
−0.976824 + 0.214045i \(0.931336\pi\)
\(744\) 1.91054e15 0.307260
\(745\) −2.36967e14 −0.0378293
\(746\) 2.09014e15 0.331217
\(747\) 7.12275e14 0.112043
\(748\) 6.09496e14 0.0951726
\(749\) −7.01795e15 −1.08783
\(750\) 9.16636e13 0.0141046
\(751\) 7.74282e14 0.118271 0.0591357 0.998250i \(-0.481166\pi\)
0.0591357 + 0.998250i \(0.481166\pi\)
\(752\) 8.12810e15 1.23251
\(753\) −1.20387e15 −0.181220
\(754\) 6.76847e14 0.101146
\(755\) −7.36018e14 −0.109189
\(756\) −1.70882e15 −0.251667
\(757\) 5.19332e15 0.759307 0.379653 0.925129i \(-0.376043\pi\)
0.379653 + 0.925129i \(0.376043\pi\)
\(758\) −5.91367e13 −0.00858374
\(759\) 4.53822e15 0.653968
\(760\) −2.15158e13 −0.00307811
\(761\) 7.14791e15 1.01523 0.507614 0.861585i \(-0.330528\pi\)
0.507614 + 0.861585i \(0.330528\pi\)
\(762\) −1.14589e15 −0.161581
\(763\) 3.58669e15 0.502122
\(764\) 6.63202e14 0.0921790
\(765\) 1.19795e13 0.00165310
\(766\) −1.77289e14 −0.0242898
\(767\) −5.75897e14 −0.0783376
\(768\) −2.83382e15 −0.382724
\(769\) 8.90844e15 1.19456 0.597278 0.802034i \(-0.296249\pi\)
0.597278 + 0.802034i \(0.296249\pi\)
\(770\) −1.60986e14 −0.0214333
\(771\) −7.46091e14 −0.0986263
\(772\) −5.32842e15 −0.699364
\(773\) −1.46312e16 −1.90674 −0.953372 0.301799i \(-0.902413\pi\)
−0.953372 + 0.301799i \(0.902413\pi\)
\(774\) 6.67502e14 0.0863729
\(775\) 1.11319e16 1.43024
\(776\) 6.06569e14 0.0773821
\(777\) −8.10711e13 −0.0102695
\(778\) −2.09643e15 −0.263689
\(779\) 8.78876e14 0.109767
\(780\) −1.75324e14 −0.0217431
\(781\) 1.91200e16 2.35455
\(782\) −1.03396e14 −0.0126434
\(783\) 1.41244e15 0.171507
\(784\) 6.22119e15 0.750128
\(785\) −5.78861e14 −0.0693093
\(786\) −5.67862e14 −0.0675179
\(787\) −6.72743e15 −0.794306 −0.397153 0.917752i \(-0.630002\pi\)
−0.397153 + 0.917752i \(0.630002\pi\)
\(788\) −2.41777e15 −0.283479
\(789\) −4.87925e15 −0.568106
\(790\) −1.05309e14 −0.0121763
\(791\) 3.45304e15 0.396489
\(792\) 1.39873e15 0.159494
\(793\) −9.49673e15 −1.07540
\(794\) −1.21743e15 −0.136909
\(795\) 1.66809e13 0.00186295
\(796\) 9.00599e15 0.998872
\(797\) 1.13524e16 1.25045 0.625224 0.780446i \(-0.285008\pi\)
0.625224 + 0.780446i \(0.285008\pi\)
\(798\) −1.72793e14 −0.0189021
\(799\) −9.69179e14 −0.105292
\(800\) 4.97688e15 0.536986
\(801\) 3.08835e15 0.330939
\(802\) −1.01957e15 −0.108507
\(803\) −6.34529e15 −0.670681
\(804\) −5.60358e15 −0.588244
\(805\) −7.40304e14 −0.0771850
\(806\) 1.57424e15 0.163015
\(807\) 6.03441e15 0.620627
\(808\) −4.54414e15 −0.464183
\(809\) 3.18993e15 0.323642 0.161821 0.986820i \(-0.448263\pi\)
0.161821 + 0.986820i \(0.448263\pi\)
\(810\) 1.34968e13 0.00136008
\(811\) 7.60095e15 0.760769 0.380385 0.924828i \(-0.375792\pi\)
0.380385 + 0.924828i \(0.375792\pi\)
\(812\) −1.17227e16 −1.16539
\(813\) 1.68704e15 0.166582
\(814\) 3.25788e13 0.00319522
\(815\) 6.08343e13 0.00592628
\(816\) 4.07882e14 0.0394674
\(817\) −1.82968e15 −0.175854
\(818\) 3.64688e15 0.348160
\(819\) −2.86798e15 −0.271966
\(820\) 5.69758e14 0.0536679
\(821\) −1.64757e16 −1.54154 −0.770771 0.637112i \(-0.780129\pi\)
−0.770771 + 0.637112i \(0.780129\pi\)
\(822\) −1.53605e15 −0.142761
\(823\) −3.76330e15 −0.347432 −0.173716 0.984796i \(-0.555577\pi\)
−0.173716 + 0.984796i \(0.555577\pi\)
\(824\) 3.54417e15 0.325024
\(825\) 8.14977e15 0.742418
\(826\) −3.67953e14 −0.0332968
\(827\) 2.15427e15 0.193651 0.0968254 0.995301i \(-0.469131\pi\)
0.0968254 + 0.995301i \(0.469131\pi\)
\(828\) 3.15782e15 0.281982
\(829\) 1.72483e16 1.53002 0.765010 0.644018i \(-0.222734\pi\)
0.765010 + 0.644018i \(0.222734\pi\)
\(830\) 4.66919e13 0.00411445
\(831\) −1.01952e16 −0.892464
\(832\) −5.48591e15 −0.477057
\(833\) −7.41804e14 −0.0640828
\(834\) −2.02052e15 −0.173401
\(835\) −1.04373e15 −0.0889844
\(836\) −1.88229e15 −0.159423
\(837\) 3.28512e15 0.276415
\(838\) −2.22929e15 −0.186348
\(839\) −2.90763e15 −0.241461 −0.120731 0.992685i \(-0.538524\pi\)
−0.120731 + 0.992685i \(0.538524\pi\)
\(840\) −2.28169e14 −0.0188244
\(841\) −2.51093e15 −0.205806
\(842\) −9.08359e14 −0.0739675
\(843\) 6.80852e15 0.550809
\(844\) −8.48109e15 −0.681661
\(845\) 5.18445e14 0.0413991
\(846\) −1.09194e15 −0.0866286
\(847\) −1.14841e16 −0.905185
\(848\) 5.67960e14 0.0444775
\(849\) −3.28711e15 −0.255754
\(850\) −1.85679e14 −0.0143535
\(851\) 1.49816e14 0.0115066
\(852\) 1.33042e16 1.01525
\(853\) 8.71618e15 0.660856 0.330428 0.943831i \(-0.392807\pi\)
0.330428 + 0.943831i \(0.392807\pi\)
\(854\) −6.06766e15 −0.457092
\(855\) −3.69959e13 −0.00276911
\(856\) 3.99714e15 0.297265
\(857\) −5.16908e15 −0.381961 −0.190980 0.981594i \(-0.561167\pi\)
−0.190980 + 0.981594i \(0.561167\pi\)
\(858\) 1.15251e15 0.0846186
\(859\) −1.66371e16 −1.21371 −0.606855 0.794813i \(-0.707569\pi\)
−0.606855 + 0.794813i \(0.707569\pi\)
\(860\) −1.18614e15 −0.0859797
\(861\) 9.32021e15 0.671287
\(862\) 3.52089e15 0.251978
\(863\) −6.79105e14 −0.0482923 −0.0241461 0.999708i \(-0.507687\pi\)
−0.0241461 + 0.999708i \(0.507687\pi\)
\(864\) 1.46872e15 0.103780
\(865\) 2.28843e14 0.0160675
\(866\) 3.49528e15 0.243856
\(867\) 8.27944e15 0.573979
\(868\) −2.72652e16 −1.87824
\(869\) −1.87655e16 −1.28455
\(870\) 9.25902e13 0.00629809
\(871\) −9.40474e15 −0.635692
\(872\) −2.04283e15 −0.137212
\(873\) 1.04298e15 0.0696140
\(874\) 3.19314e14 0.0211790
\(875\) −2.66451e15 −0.175619
\(876\) −4.41523e15 −0.289188
\(877\) 8.88982e15 0.578623 0.289311 0.957235i \(-0.406574\pi\)
0.289311 + 0.957235i \(0.406574\pi\)
\(878\) −2.18431e15 −0.141284
\(879\) −1.12796e16 −0.725025
\(880\) −1.17358e15 −0.0749647
\(881\) 2.92895e16 1.85928 0.929640 0.368469i \(-0.120118\pi\)
0.929640 + 0.368469i \(0.120118\pi\)
\(882\) −8.35763e14 −0.0527237
\(883\) −1.75301e16 −1.09901 −0.549504 0.835491i \(-0.685183\pi\)
−0.549504 + 0.835491i \(0.685183\pi\)
\(884\) 7.11793e14 0.0443472
\(885\) −7.87806e13 −0.00487788
\(886\) 1.16535e15 0.0717086
\(887\) 1.36741e16 0.836215 0.418107 0.908398i \(-0.362694\pi\)
0.418107 + 0.908398i \(0.362694\pi\)
\(888\) 4.61748e13 0.00280630
\(889\) 3.33091e16 2.01189
\(890\) 2.02451e14 0.0121528
\(891\) 2.40507e15 0.143483
\(892\) 8.26417e15 0.489996
\(893\) 2.99309e15 0.176375
\(894\) 1.08391e15 0.0634801
\(895\) −1.11238e15 −0.0647479
\(896\) −1.61446e16 −0.933968
\(897\) 5.29991e15 0.304727
\(898\) 3.28057e15 0.187469
\(899\) 2.25364e16 1.27999
\(900\) 5.67083e15 0.320120
\(901\) −6.77225e13 −0.00379968
\(902\) −3.74537e15 −0.208862
\(903\) −1.94032e16 −1.07545
\(904\) −1.96671e15 −0.108346
\(905\) −3.52909e14 −0.0193239
\(906\) 3.36662e15 0.183227
\(907\) −9.52788e15 −0.515414 −0.257707 0.966223i \(-0.582967\pi\)
−0.257707 + 0.966223i \(0.582967\pi\)
\(908\) 1.65216e16 0.888342
\(909\) −7.81351e15 −0.417585
\(910\) −1.88005e14 −0.00998717
\(911\) 3.41678e16 1.80412 0.902062 0.431607i \(-0.142053\pi\)
0.902062 + 0.431607i \(0.142053\pi\)
\(912\) −1.25965e15 −0.0661117
\(913\) 8.32027e15 0.434058
\(914\) −2.38774e15 −0.123818
\(915\) −1.29912e15 −0.0669626
\(916\) 1.28309e16 0.657401
\(917\) 1.65068e16 0.840682
\(918\) −5.47953e13 −0.00277401
\(919\) 4.64389e15 0.233693 0.116847 0.993150i \(-0.462721\pi\)
0.116847 + 0.993150i \(0.462721\pi\)
\(920\) 4.21647e14 0.0210919
\(921\) 1.36122e16 0.676865
\(922\) −6.76212e15 −0.334243
\(923\) 2.23291e16 1.09714
\(924\) −1.99611e16 −0.974965
\(925\) 2.69041e14 0.0130629
\(926\) −5.74613e15 −0.277342
\(927\) 6.09410e15 0.292396
\(928\) 1.00756e16 0.480572
\(929\) −3.16080e16 −1.49869 −0.749343 0.662182i \(-0.769631\pi\)
−0.749343 + 0.662182i \(0.769631\pi\)
\(930\) 2.15350e14 0.0101505
\(931\) 2.29089e15 0.107345
\(932\) −3.05287e16 −1.42207
\(933\) 1.33634e16 0.618825
\(934\) −2.05383e15 −0.0945488
\(935\) 1.39935e14 0.00640417
\(936\) 1.63349e15 0.0743187
\(937\) 1.75268e16 0.792747 0.396373 0.918089i \(-0.370269\pi\)
0.396373 + 0.918089i \(0.370269\pi\)
\(938\) −6.00889e15 −0.270196
\(939\) −8.84441e15 −0.395374
\(940\) 1.94036e15 0.0862342
\(941\) 2.85640e16 1.26205 0.631025 0.775763i \(-0.282635\pi\)
0.631025 + 0.775763i \(0.282635\pi\)
\(942\) 2.64777e15 0.116306
\(943\) −1.72234e16 −0.752149
\(944\) −2.68236e15 −0.116458
\(945\) −3.92330e14 −0.0169346
\(946\) 7.79726e15 0.334611
\(947\) 2.69380e16 1.14932 0.574659 0.818393i \(-0.305135\pi\)
0.574659 + 0.818393i \(0.305135\pi\)
\(948\) −1.30576e16 −0.553881
\(949\) −7.41028e15 −0.312515
\(950\) 5.73426e14 0.0240435
\(951\) −2.40562e15 −0.100285
\(952\) 9.26336e14 0.0383942
\(953\) 1.64423e16 0.677567 0.338783 0.940864i \(-0.389985\pi\)
0.338783 + 0.940864i \(0.389985\pi\)
\(954\) −7.63004e13 −0.00312616
\(955\) 1.52265e14 0.00620273
\(956\) −3.04745e16 −1.23429
\(957\) 1.64991e16 0.664423
\(958\) 7.77654e15 0.311369
\(959\) 4.46503e16 1.77755
\(960\) −7.50452e14 −0.0297051
\(961\) 2.70076e16 1.06294
\(962\) 3.80468e13 0.00148887
\(963\) 6.87296e15 0.267423
\(964\) 5.24972e14 0.0203101
\(965\) −1.22336e15 −0.0470603
\(966\) 3.38623e15 0.129522
\(967\) −1.15362e16 −0.438749 −0.219375 0.975641i \(-0.570402\pi\)
−0.219375 + 0.975641i \(0.570402\pi\)
\(968\) 6.54086e15 0.247355
\(969\) 1.50198e14 0.00564787
\(970\) 6.83704e13 0.00255637
\(971\) 1.06005e16 0.394114 0.197057 0.980392i \(-0.436862\pi\)
0.197057 + 0.980392i \(0.436862\pi\)
\(972\) 1.67351e15 0.0618677
\(973\) 5.87332e16 2.15905
\(974\) −3.32971e15 −0.121712
\(975\) 9.51762e15 0.345942
\(976\) −4.42329e16 −1.59872
\(977\) −3.28987e16 −1.18239 −0.591193 0.806530i \(-0.701343\pi\)
−0.591193 + 0.806530i \(0.701343\pi\)
\(978\) −2.78263e14 −0.00994469
\(979\) 3.60758e16 1.28207
\(980\) 1.48514e15 0.0524837
\(981\) −3.51259e15 −0.123438
\(982\) −4.52313e15 −0.158062
\(983\) 4.29176e16 1.49139 0.745695 0.666287i \(-0.232118\pi\)
0.745695 + 0.666287i \(0.232118\pi\)
\(984\) −5.30841e15 −0.183439
\(985\) −5.55099e14 −0.0190753
\(986\) −3.75904e14 −0.0128456
\(987\) 3.17408e16 1.07863
\(988\) −2.19821e15 −0.0742859
\(989\) 3.58563e16 1.20500
\(990\) 1.57660e14 0.00526899
\(991\) 2.56509e16 0.852507 0.426254 0.904604i \(-0.359833\pi\)
0.426254 + 0.904604i \(0.359833\pi\)
\(992\) 2.34343e16 0.774531
\(993\) −1.84021e16 −0.604847
\(994\) 1.42665e16 0.466330
\(995\) 2.06770e15 0.0672141
\(996\) 5.78947e15 0.187160
\(997\) −4.97940e16 −1.60086 −0.800430 0.599426i \(-0.795396\pi\)
−0.800430 + 0.599426i \(0.795396\pi\)
\(998\) −4.29742e14 −0.0137401
\(999\) 7.93961e13 0.00252458
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.12.a.c.1.13 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.c.1.13 27 1.1 even 1 trivial